On Success runs of a xed length dened on a q-sequence of binary trials Jungtaek Oh Dae-Gyu Jang

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On Success runs of a ﬁxed length deﬁned on a

q-sequence of binary trials

Jungtaek Oh, Dae-Gyu Jang

October 11, 2022

Abstract

We study the exact distributions of runs of a ﬁxed length in variation which

considers binary trials for which the probability of ones is geometrically varying.

The random variable En,k denote the number of success runs of a ﬁxed length k,

1≤k≤n. Theorem 3.1 gives an closed expression for the probability mass function

(PMF) of the TypeIV q-binomial distribution of order k. Theorem 3.2 and Corollary

3.1 gives an recursive expression for the probability mass function (PMF) of the

TypeIV q-binomial distribution of order k. The probability generating function and

moments of random variable En,k are obtained as a recursive expression. We address

the parameter estimation in the distribution of En,k by numerical techniques. In the

present work, we consider a sequence of independent binary zero and one trials with

not necessarily identical distribution with the probability of ones varying according

to a geometric rule. Exact and recursive formulae for the distribution obtained by

means of enumerative combinatorics.

Contents

1 Introduction 2

2 Preliminary and Notation 4

3 Type IV q-binomial distribution of order k6

3.1 PGF, MGF and moments of En,k ....................... 11

3.2 Closed formulae for the mean and variance of En,k .............. 14

arXiv:2210.04521v1 [math.PR] 10 Oct 2022

[1110][11110][11110][0][110]

𝜃𝑞1-𝜃𝑞𝜃𝑞1-𝜃𝑞𝜃𝑞1-𝜃𝑞1-𝜃𝑞𝜃𝑞1-𝜃𝑞

4 Simulation Study 18

1 Introduction

Charalambides (2010b) studied discrete q-distributions on Bernoulli trials with a geomet-

rically varying success probability. Let us consider a sequence X1,...,Xnof zero(failure)-

one(success) Bernoulli trials, such that the trials of the subsequence after the (i−1)st

zero until the ith zero are independent with equal failure probability. The i’s geometric

sequences of trials is the subsequence after the (i−1)’st zero and until the i’th zero, for

i > 0 and the subsequence after the (j−1)’st zero and until the j’th zero, for j > 0 are

independent for all i6=j(i.e. i’th and j’th geometric sequences are independent) with

probability of zeros at the ith geometric sequence of trials

qi= 1 −θqi−1, i = 1,2, ..., 0≤θ≤1,0≤q < 1.(1.1)

We note that probability of failures in the independent geometric sequences of trials is

geometrically increasing with rate q. Let S(0)

j= Σj

m=1(1 −Xm) denote the number of zeros

in the ﬁrst jtrials. Because the probability of zero’s at the ith geometric sequence of

trials is in fact the conditional probability of occurrence of a zero at any trial jgiven the

occurrence of i−1 zeros in the previous trials. We can rewrite as follows.

qj,i =pXj= 0 S(0)

j−1=i−1= 1 −θqi−1, i = 1,2, ..., j, j = 1,2, .... (1.2)

We note that (1.1) is exactly the conditional probability in (1.2). To make more clear

and transparent the preceding, we consider an example n= 18, the binary sequence

111011110111100110, each subsequence has own success and failure probabilities according

to a geometric rule.

This stochastic model (1.1) or (1.2) has interesting applications, studied as a reliabil-

ity growth model by Dubman and Sherman (1969), and applies to a q-boson theory in

physics by Jing and Fan (1994) and Jing (1994). More speciﬁcally, q-binomial distribution

introduced as a q-deformed binomial distribution, in order to set up a q-binomial state.

This stochastic model (1.1) also applies to start-up demonstration tests, as a sequential-

intervention model which is proposed by Balakrishnan et al. (1995).

The stochastic model (1.1) is q-analogue of the classical binomial distribution with

geometrically varying probability of zeros, which is a stochastic model of an independent

and identically distributed (IID) trials with failure probability is

πj=P(Xj= 0) = 1 −θ, j = 1,2,..., 0< θ < 1.(1.3)

As qtends toward 1, the stochastic model (1.1) reduces to IID(Bernoulli) model (1.3), since

qi→πi,i= 1,2, . . . or qj,i →1−θ,i= 1,2, . . . , j,j= 1,2,....

The Discrete q-distributions based on the stochastic model of the sequence of indepen-

dent Bernoulli trials have been investigated by numerous researchers, for a lucid review

and comprehensive list of publications on this area the interested reader may consult the

monographs by Charalambides (2010b,a, 2016).

From a Mathematical and Statistical point of view, Charalambides (2016) mentioned

the preface of his book ”It should be noticed that a stochastic model of a sequence of in-

dependent Bernoulli trials, in which the probability of success at a trial is assumed to vary

with the number of trials and/or the number of successes, is advantageous in the sense that

it permits incorporating the experience gained from previous trials and/or successes. If the

probability of success at a trial is a very general function of the number of trials and/or the

number successes, very little can be inferred from it about the distributions of the various

random variables that may be deﬁned on this model. The assumption that the probability

of success (or failure) at a trial varies geometrically, with rate (proportion) q, leads to the

introduction of discrete q-distributions”.

The distribution theory of runs and patterns has been incredibly developed in the last

few decades through a slew of the research literature because of their theoretical interest

and applications in a wide variety of research areas such as hypothesis testing, system reli-

ability, quality control, physics, psychology, radar astronomy, molecular biology, computer

science, insurance, and ﬁnance. During the past few decades up to recently, the meaningful

progress on runs and pattern statistics has been wonderfully surveyed in Balakrishnan and

Koutras (2003) as well as in Fu and Lou (2003) and references therein. Furthermore, there

are some more recent contributions on the topic such as Arapis et al. (2018), Eryilmaz

(2018), Kong (2019), Makri et al. (2019), and Aki (2019).

There are several ways of counting scheme. Each counting scheme depends on diﬀerent

conditions: whether or not the overlapping counting is permitted, and whether or not the

counting starts from scratch when a certain kind or size of run has been so far enumer-

ated. Feller (1968)Feller (1968) proposed a classical counting method, once kconsecutive

successes show up, the number of occurrences of kconsecutive successes is counted and the

counting procedure starts anew, called non-overlapping counting scheme which is referred

to as Type Idistributions of order k. A second scheme can be initiated by counting a

success runs of length greater than or equal to kpreceded and followed by a failure or by

the beginning or by the end of the sequence (see. e.g. Mood, 1940 or Gibbons, 1971 or

Goldstein, 1990) and is usually called at least counting scheme which is referred to as Type

II distributions of order k. Ling (1988) suggested the overlapping counting scheme, an

uninterrupted sequence of m≥ksuccesses preceded and followed by a failure or by the

beginning or by the end of the sequence. It accounts for m−k+ 1 success runs of length

of kwhich is referred to as Type III distributions of order k. Mood(1940) suggested exact

counting scheme, asuccess run of length exactly kpreceded and succeeded by failure or by

nothing which is referred to as Type IV distributions of order k.

According to the three aforementioned counting schemes,the random variables of the

number of runs of length kcounted in noutcomes, have three diﬀerent distributions which

are denoted as Nn,k,Gn,k,Mn,k and En,k. Moreover, if the underline sequence is an inde-

pendent and identically distributed (i.i.d.) sequence of random variables, X1, X2, . . . , Xn,

then distributions of Nn,k,Gn,k,Mn,k and En,k will be referred to as Type I,II,III and

IV binomial distributions of order k.

To make more clear the distinction between the aforementioned counting methods we

mention by way of example that for n= 12, the binary sequence 011111000111 contains

N12,2= 3, G12,2= 2, M12,2= 6, T(I)

2,2= 5, T(II)

2,2= 11, and T(III)

2,2= 4.

2 Preliminary and Notation

We ﬁrst recall some deﬁnitions, notation and known results in which will be used in this

paper. Throughout the paper, we suppose that 0 < q < 1. First, we introduce the following

notation.

•L(1)

n: the length of the longest run of successes in X1, X2, . . . , Xn;

•L(0)

n: the length of the longest run of failures in X1, X2, . . . , Xn;

•Sn: the total number of successes in X1, X2, . . . , Xn;

•Fn: the total number of failures in X1, X2, . . . , Xn.

Next, let us introduce some basic q-sequences and functions and their properties, which are

useful in the sequel. The q-shifted factorials are deﬁned as

(a;q)0= 1,(a;q)n=

n−1

k=0

(1 −aqk),(a;q)∞=

∞

k=0

(1 −aqk).(2.1)

Let m,nand ibe positive integer and zand qbe real numbers, with q6= 1. The number

[z]q= (1 −qz)/(1 −q) is called q-number and in particular [z]qis called q-integer. The m

th order factorial of the q-number [z]q, which is deﬁned by

[z]m,q =

i=1

[z−i+ 1]q= [z]q[z−1]q· · · [z−m+ 1]q

=(1 −qz)(1 −qz−1)· · · (1 −qz−m+1)

(1 −q)m, z = 1,2, . . . , m = 0,1, . . . , z.

(2.2)

is called q-factorial of zof order m. In particular, [m]q! = [1]q[2]q...[m]qis called q-factorial

of m. The q-binomial coeﬃcient (or Gaussian polynomial) is deﬁned by

n

mq

=[n]m,q

[m]q!=[n]q!

[m]q![n−m]q!=(1 −qn)(1 −qn−1)· · · (1 −qn−m+1)

(1 −qm)(1 −qm−1)· · · (1 −q)

=(q;q)n

(q;q)m(q;q)n−m

, m = 1,2,...,

(2.3)

The q-binomial (q-Newton’s binomial) formula is expressed as

i=1

(1 + zqi−1) =

k=0

qk(k−1)/2n

kq

zk,−∞ < z < ∞, n = 1,2,.... (2.4)

For q→1 the q-analogs tend to their classical counterparts, that is

lim

q→1n

rq

=n

r

Let us consider again a sequence of independent geometric sequences of trials with prob-

ability of failure at the ith geometric sequence of trials given by (1.1) or (1.2). We are

interesting now is focused on the study of the number of successes in a given number of

trials in this stochastic model.

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OnSuccessrunsofaxedlengthdenedonaq-sequenceofbinarytrialsJungtaekOh,Dae-GyuJangOctober11,2022AbstractWestudytheexactdistributionsofrunsofaxedlengthinvariationwhichconsidersbinarytrialsforwhichtheprobabilityofonesisgeometricallyvarying.TherandomvariableEn;kdenotethenumberofsuccessrunsofaxedlength...

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On Success runs of a xed length dened on a q-sequence of binary trials Jungtaek Oh Dae-Gyu Jang

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